Showing papers on "Elementary function published in 2005"

High-speed function approximation using a minimax quadratic interpolator

Abstract: A table-based method for high-speed function approximation in single-precision floating-point format is presented in this paper. Our focus is the approximation of reciprocal, square root, square root reciprocal, exponentials, logarithms, trigonometric functions, powering (with a fixed exponent p), or special functions. The algorithm presented here combines table look-up, an enhanced minimax quadratic approximation, and an efficient evaluation of the second-degree polynomial (using a specialized squaring unit, redundant arithmetic, and multioperand addition). The execution times and area costs of an architecture implementing our method are estimated, showing the achievement of the fast execution times of linear approximation methods and the reduced area requirements of other second-degree interpolation algorithms. Moreover, the use of an enhanced minimax approximation which, through an iterative process, takes into account the effect of rounding the polynomial coefficients to a finite size allows for a further reduction in the size of the look-up tables to be used, making our method very suitable for the implementation of an elementary function generator in state-of-the-art DSPs or graphics processing units (GPUs).

Incomplete beta-function expansions of the solutions to the confluent Heun equation

Abstract: Several expansions of the solutions to the confluent Heun equation in terms of incomplete beta functions are constructed. A new type of expansion involving certain combinations of the incomplete beta functions as expansion functions is introduced. The necessary and sufficient conditions when the derived expansions are terminated, thus generating closed-form solutions, are discussed. It is shown that termination of a beta-function series solution always leads to a solution that is necessarily an elementary function.

Kober fractional q-integral operator of the basic analogue of the H-function

Abstract: This paper deals with the derivation of the Kober fractional q-integral operator of the basic analogue of the H-function defined by Saxena, Modi and Kalla [Rev. Tec. Ing., Univ. Zulia. 6(1983), 139-143]. Several interesting results .involving Gq(.);Eq(.); the basic elementary functions and the basic Bessel functions such as Jv(x; q); Yv(x; q); Kv(x; q); Hv(x; q), are deduced as the special cases of the main results.

New Results on the Distance between a Segment and Z². Application to the Exact Rounding

Abstract: This paper presents extensions to Lefevre's algorithm that computes a lower bound on the distance between a segment and a regular grid Zopf2. This algorithm and, in particular, the extensions are useful in the search for worst cases for the exact rounding of unary elementary functions or base-conversion functions. The proof that is presented is simpler and less technical than the original proof. This paper also gives benchmark results with various optimization parameters, explanations of these results, and an application to base conversion

A simple approximation of the Maliuzhinets function for describing wedge diffraction

Abstract: A new approximation of the Maliuzhinets function /spl psi//sub /spl Phi//(/spl alpha/) by a single cosine function is proposed. This is the simplest approximation of the function available in the literature. The use of the cosine approximation permits representing fields scattered by nonmetallic wedges in terms of elementary functions only. The accuracy of the approximation, as applied to the case of an impedance wedge under normal illumination, is estimated both analytically and numerically.

A New Elementary Function for Our Curricula

Abstract: The concept of function is central to the teaching and learning of mathematics. Indeed it has been variously said that the single most important concept in modern mathematics is that of the function. This article explains this new function and also states how to use it in the calculus. As a mathematical notion, the concept of function is fundamental, yet powerful, and is a unifying theme that is found running throughout most branches of mathematics. One particular area where the concept of function finds its raison d' tre is in the so-called special functions. The special functions are those functions that arise most frequently in applications and have been studied and used for centuries. In this paper the author introduces one of these special functions, the Lambert W function.

Interval Mathematical Library Based on Chebyshev and Taylor Series Expansion

Abstract: In this paper, we present a mathematical library designed for use in interval solvers of nonlinear systems of equations. The library computes the validated upper and lower bounds of ranges of values of elementary mathematical functions on an interval, which are optimal in most cases. Computation of elementary functions is based on their expansion in Chebyshev and Taylor series and uses the rounded directions setting mechanism. Some original techniques developed by the authors are applied in order to provide high speed and accuracy of the computation.

Factorization of Block Triangular Matrix Functions with Off-diagonal Binomials

Abstract: Factorizations of Wiener-Hopf type are considered in the abstract framework of Wiener algebras of matrix-valued functions on connected compact abelian groups, with a non-archimedean linear order on the dual group. A criterion for factorizability is established for 2 × 2 block triangular matrix functions with elementary functions on the main diagonal and a binomial expression in the off-diagonal block.

Calculation of potential flows

Abstract: Methods for calculating potential flows based on the theory of functions of a complex variable have been widely used as fundamental investigation methods in many fields of engineering, including seepage theory, elasticity theory, continuum mechanics, heat dynamics, aero- and hydromechanics, electromagnetism, electroand radio engineering, etc. [1‐3]. In most cases, the application of these methods involves conformal mapping of the rectangle 1 — 2 — 3 — 4 of the complex domain W = ϕ + i ψ (Fig. 1a) onto a complex half-plane ζ = ξ + i η (Fig. 1b). It is well known [1‐5] that this mapping is performed by means of Jacobi elliptic functions, making use of the complete elliptic integrals of the first kind K and K ’ (Fig. 1) with the modulus λ and the complementary modulus λ ’ = , respectively. This generates considerable difficulties due to the necessity of series expansion of elliptic functions, interpolation of special nonograms and tables, solution of inverse table problems, etc., particularly when it comes to determining the current values of Jacobi functions for the rectangle interior [1‐4, 6, 7]. Moreover, the difficulty of expressing the elliptic functions in terms of elementary functions restricts the possibility of analytically representing the relationship between the physical parameters of the problem under consideration and the given boundary conditions, as well as the use of complicated calculation techniques. The above circumstances considerably constrain the further development of analytical methods for investigating engineering problems in the above-listed lines of inquiry. In this study, we present a new method for solving this problem based on the conformal mapping of a rectangle, one side of which has a vanishingly small convexity, onto a half-plane by means of elementary functions. For this purpose, in the complex half-band W = ϕ + i ψ of width H (Fig. 2), we introduce the function

The Mathematics Companion: Mathematical Methods for Physicists and Engineers

Abstract: Part 1 Essential Mathematics: Basic mathematics. Differentiation. Integration. Exponentials and logarithms. Hyperbolic functions. Infinite series. Part 2 Advance Mathematics: Ordinary differential equations. Laplace transforms. Vector analysis. Partial derivatives. Multiple integrals. Fourier series. Special functions. Partial differential equations.

Hardware Algorithms for Division, Square Root and Elementary Functions

Abstract: Many numerically intensive applications require the fast computation of division, square root and elementary functions. Graphics processing, image processing and generally digital signal processing are examples of such applications. Motivated by this demand on high performance computation many algorithms have been proposed to carry out the computation task in hardware instead of software. The reason behind this migration is to increase the performance of the algorithms since more than an order of magnitude increase in performance can be attained by such migration. The hardware algorithms represent families of algorithms that cover a wide spectrum of speed and cost. This thesis presents a survey about the hardware algorithms for the computation of elementary functions, division and square root. Before we present the different algorithms we discuss argument reduction techniques, an important step in the computation task. We then present the approximation algorithms. We present polynomial based algorithms, table and add algorithms, a powering algorithm, functional recurrence algorithms used for division and square root and two digit recurrence algorithms namely the CORDIC and the Briggs and DeLugish algorithm. Careful error analysis is crucial not only for correct algorithms but it may also lead to better circuits from the point of view of area, delay or power. Error analysis of the different algorithms is presented. We made three contributions in this thesis. The first contribution is an algorithm that computes a truncated version of the minimax polynomial coefficients that gives better results than the direct rounding. The second contribution is about devising an algorithmic error analysis that proved to be more accurate and led to a substantial decrease in the area of the tables used in a powering, division iii and square root algorithms. Finally the third contribution is a proposed high order Newton-Raphson algorithm for the square root reciprocal operation and a square root circuit based on this algorithm. VHDL models for the powering algorithm, functional recurrence algorithms and the CORDIC algorithm are developed to verify these algorithms. Behavioral simulation is also carried out with more than two million test vectors for the powering algorithm and the functional recurrence algorithms. The models passed all the tests.

Three-particle correlation function in the electron-plasmon model

Abstract: The method of calculating the n-particle electron correlation functions for the electron-plasmon model is demonstrated. We have proposed this model earlier for the description of the strongly non-ideal electron liquid. The three-particle dynamic correlation function is calculated and presented in the elementary functions. The differences from the similar correlation function of the ordinary reference system approach in the electron liquid theory are investigated.

New anisotropic models from isotropic solutions

Abstract: We establish an algorithm that produces a new solution to the Einstein field equations, with an anisotropic matter distribution, from a given seed isotropic solution. The new solution is expressed in terms of integrals of known functions, and the integration can be completed in principle. The applicability of this technique is demonstrated by generating anisotropic isothermal spheres and anisotropic constant density Schwarzschild spheres. Both of these solutions are expressed in closed form in terms of elementary functions, and this facilitates physical analysis.